in reference to the Phoneidoscope. 81 



where 



2s = h + k + l. 



This expression vanishes when 



. 9 p r s(s—7i) . 9 cf s(s — P) . „r' s(s — l) 

 Sm 2 = kl> Sln 2 = V-' Sm 2 = V * 



These values show that there is a node in each triangle the 

 position of which depends on the relative amplitudes of the 

 waves. If one of the amplitudes (say I) is very nearly equal 

 to the sum of the other two, (s — P) is very small, and there- 

 fore r 7 is very small, and consequently r is very small; 

 the result is that the nodes lie in pairs very near together. 

 When one amplitude equals the sum of the other two, the 

 pairs of nodes coalesce ; and when one amplitude is greater 

 than the sum of the other two, there can be no node. 



In the particular case in which the directions of the waves 

 are inclined at angles of 120° to one another, and the ampli- 

 tudes are equal, the triangles in fig. 2 become equilateral, with 



2\ 



their sides = -^-, and the nodes are equidistant from the ven- 

 tral segments. Also the difference of phase between succes- 

 sive angles of the triangles is 120°. In fig. 3 let the dots 

 represent the nodes and the numbers the ventral segments. 

 Then all the ones move together, and so do all the twos, and 

 also all the threes, the difference of phase between the different 

 sets being 120°. 



This last case may be investigated algebraically. Let h = 

 the amplitude of each wave, v the wave-velocity, t the time, 

 x, y and r, 6 the coordinates of a point in the film, and z the 

 displacement at that point, at the time t. Take a ventral seg- 

 ment as origin, and the direction of one of the waves as axis 

 of x. Then 



h~ l z= cos [2tt\- 1 ^vt— r cos 0J-] 



+ cos [27TX- 1 [vt-r cos (0- 120°)} ] 

 + cos [277-A,- 1 \vt-r cos ((9 + 120°)} ] 

 = cos V27r\- l (yt— x)l 



+ 2 cos J 2tt\- 1 A;*+!) |cos(w\- , y > /3). 



When z is a maximum we must have 

 cos \27r\- l (vt—x)\=l; 



